Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

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2.4 Inhomogeneous Particles

111

The Stratton–Chu representation theorem for the scattered field Es in Ds, yields the expansion of the approximate scattered field ENs in the exterior of a sphere enclosing the particle

N

ENs (r1) = fνN M 3ν (ksr1) + gνN N 3ν (ksr1) ,

 

 

 

 

ν=1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where the expansion coe cients are given by

 

 

 

 

$ fνN %

 

jks2

 

N

 

 

$ N

1

 

(ksr1) %

 

 

 

 

 

 

ν

 

 

gN

=

 

 

 

 

 

 

 

ei,1 (r ) ·

 

 

 

 

1

 

(ksr )

 

 

 

π S

 

M

 

 

 

 

 

1

ν

 

 

 

ν

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

$ M

1

 

(ksr1) %

 

 

 

 

 

 

 

µs

N

 

 

 

 

 

 

 

 

ν

dS (r

 

 

 

+ j

 

 

h

(r )

·

 

 

 

 

 

 

 

 

 

) .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

εs

 

i,1

 

N

1

(ksr )

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ν

1

 

 

Taking into account the expressions of the approximate surface fields given by (2.73), we derive the matrix equation

s = Q111 (ks, ki,1) i1 + Q113 (ks, ki,1) i1 ,

(2.79)

where s = [f N , gN ]T

is the vector containing the expansion coe cients of

ν ν

 

 

the scattered field. Combining (2.78) and (2.79) we see that the transition matrix relating the scattered field coe cients to the incident field coe cients, s = T e, is given by

1

T = − Q111 (ks, ki,1) + Q131 (ks, ki,1) T 2 Q311 (ks, ki,1) + Q331 (ks, ki,1)T 2 ,

where

(2.80)

 

T 2 = S12tr T 2S21rt .

(2.81)

For a homogeneous particle T 2 = 0, and we obtain the result established in Sect. 2.1

T = −Q111 (ks, ki) Q311 (ks, ki) 1 .

The expression of the transition matrix can also be written as

T =

T 1

Q13

(ks, ki,1) T

2

Q31

(ks, ki,1) 1

 

 

 

 

1

 

 

 

1

 

 

 

 

 

 

 

 

 

33

(ks, ki,1)T

 

Q

31

(ks, ki,1)

1

1

(2.82)

 

×

I + Q

1

2

1

 

,

 

 

 

 

 

 

 

 

 

 

 

where

T 1 = −Q111 (ks, ki) Q311 (ks, ki) 1

is the transition matrix of the host particle. If the coordinate systems O1x1y1z1 and O2x2y2z2 coincide, (2.82) is identical to the result given by Peterson and


112 2 Null-Field Method

Str¨om [189]. The various terms obtained by a formal expansion of the inverse in (2.82) can be interpreted as various multiple-scattering contributions to the total transition matrix. Indeed, using the representation

T= T 1 − Q131 T 2 Q311 1 I − Q331 T 2 Q311 1 + ...

=T 1 − Q131 T 2 Q311 1 − T 1Q331 T 2 Q311 1 +Q131 T 2 Q311 1 Q331 T 2 Q311 1 + . . .

we see that the term T 1 represents a reflection at S1, Q131 T 2(Q311 )1 represents a passage of a wave through S1 and a reflection at S2, T 1Q331 T 2(Q311 )1 represents a refraction of a wave through S1 and two consecutive reflections at S2 and S1, etc.

The expressions of the total transition matrix given by (2.80) or (2.82) are important in practical applications. As it has been shown by Peterson and Str¨om [189], this result can be extended to the case of S1 containing an arbitrary number of separate enclosures by simply replacing T 2 with the system transition matrix of the particles. In the later sections we will derive the transition matrix for a system of particles and the present formalism will enable us to analyze scattering by an arbitrarily shaped, inhomogeneous particle with an arbitrary number of irregular inclusions. In this context it should be mentioned that the separation of variables solution for a single sphere (the Lorenz–Mie theory) can be extended to spheres with one or more eccentrically positioned spherical inclusions by using the translation addition theorem for vector spherical wave functions. Theories of scattering by eccentrically stratified spheres have been derived by Fikioris and Uzunoglu [64], Borghese et al. [22], Fuller [73], Mackowski and Jones [152] and Ngo et al. [180], while treatments for a sphere with multiple spherical inclusions have been rendered by Borghese et al. [20], Fuller [75] and Ioannidou and Chrissoulidis [107]. A detailed review of the separation of variable method for inhomogeneous spheres has been given by Fuller and Mackowski [77]. The separation of variable method has also been employed in spheroidal coordinate systems by Cooray and Ciric [41] and Li et al. [141] to analyze the scattering by inhomogeneous spheroids.

2.4.2 Formulation without Addition Theorem

In our previous analysis we assumed the geometric constraint r1 > r12 (cf. (2.77)), which originates in the use of the translation addition theorem for radiating vector spherical wave functions. In this section we present a formalism that avoids the use of any local origin translation. To simplify our analysis, we assume that the coordinate systems O1x1y1z1 and O2x2y2z2 have the

same spatial orientation and set α = β = γ = 0. We begin by defining the Qpqt (k1, i, k2, j) matrix


2.4 Inhomogeneous Particles

113

 

(Qpq )11

(Qpq )12

 

 

Qtpq

(k1, i, k2, j) =

t

νµ

 

t

 

νµ

,

(2.83)

pq

21

(Q

pq

)

22

 

 

(Q )

νµ

t

νµ

 

 

 

 

t

 

 

 

 

as

(Qpqt )11νµ =

(Qpqt )12νµ =

(Qpqt )21νµ =

and

(Qpqt )22νµ =

 

jk12

 

 

 

n

r

 

 

M q k2r

 

·

 

N

p

(k1r )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

π

 

St

 

 

j

×

 

 

 

µ

j

 

 

 

 

 

 

 

ν

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"

 

 

 

ε2

 

 

 

 

 

 

q

 

 

 

 

 

 

 

p

 

 

 

 

 

 

+

 

 

 

 

 

 

 

 

 

n r

×

N

 

 

k2r

 

M

 

 

 

 

 

(k1r )

dS (r ) , (2.84)

 

 

 

 

ε1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j

 

 

µ

j

 

·

 

 

 

 

ν

 

 

 

 

 

i

 

 

jk12

 

 

 

n

r

 

N q k2r

 

 

N

p

 

(k1r )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

π

 

St

 

 

j

×

 

 

µ j

·

 

 

 

 

 

ν

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(k1r )" dS (r ) , (2.85)

+

ε2

n r

×

M q k2r

 

M

p

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ε1

 

 

j

 

 

 

µ

j

 

·

 

 

 

 

 

ν

 

 

 

 

 

i

 

 

 

jk12

 

 

n r

 

 

M q k2r

·

M

p

(k1r )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

π

 

 

St

 

 

j

×

 

 

 

µ

 

j

 

 

 

 

 

 

 

 

ν

i

 

 

 

 

 

 

 

 

 

(k1r )" dS (r ) , (2.86)

 

+

 

ε2

n r

×

N q k2r

 

N

p

 

 

 

ε1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j

 

 

 

µ

j

 

·

 

 

 

 

ν

 

 

 

 

 

i

 

 

jk12

 

n

r

 

 

N q k2r

·

 

M

p

(k1r )

 

 

 

 

 

 

 

 

π

 

St

 

 

j

×

 

 

µ

j

 

 

 

 

 

 

 

ν

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"

 

 

 

ε2

 

 

 

 

 

 

 

q

 

 

 

 

 

 

 

p

 

 

 

 

 

 

+

 

 

 

 

 

 

 

 

 

n r

×

M

 

k2r

 

N

 

 

 

 

 

(k1r )

dS (r ) . (2.87)

 

 

 

 

 

ε1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j

 

 

 

µ

j

·

 

 

 

 

ν

 

 

 

 

 

i

 

The vectors r , ri and rj are the position vectors of a point M on the surface St, and are defined with respect to the origins O, Oi and Oj , respectively (Fig. 2.3). In terms of the new Qpqt matrices, the system of matrix equations (2.75) can be written as

Q133(ks, 1, ki,1, 1)i1 + Q131(ks, 1, ki,1, 1)i1 = −e ,

 

−i1 + Q211(ki,1, 1, ki,2, 2)i2 = 0 ,

 

−Q131(ki,1, 2, ki,1, 1)i1 + Q231(ki,1, 2, ki,2, 2)i2 = 0 ,

(2.88)

while the matrix equation (2.79) reads as

 

s = Q113 (ks, 1, ki,1, 1) i1 + Q111 (ks, 1, ki,1, 1) i1 .

(2.89)

Using the substitution method we eliminate the unknown vector i2 from the last two matrix equations in (2.88) and obtain


114 2 Null-Field Method

 

M

 

ri

r

St

j

 

 

 

 

r

 

Oi

 

Oj

O

 

Fig. 2.3. Position vectors of a point M on the surface St O, Oi and Oj

i1 = T 2i1

with

with respect to the origins

(2.90)

T

2

= Q11

(ki,1, 1, ki,2, 2)

Q31

(ki,1, 2, ki,2, 2) 1

Q31

(ki,1, 2, ki,1, 1) . (2.91)

 

 

2

 

2

 

1

 

Equation (2.89) and the first matrix equation in (2.88), written in compact matrix notation as

 

s

 

 

1 (ks, ki,1)

i

 

 

 

(2.92)

 

e

=

Q

1

,

 

 

 

 

 

i1

 

 

 

 

where

 

 

 

 

 

 

 

 

 

Q1 (ks, ki,1) =

Q113 (ks, 1, ki,1, 1) Q111 (ks, 1, ki,1, 1)

,

(2.93)

−Q133(ks, 1, ki,1, 1) −Q131(ks, 1, ki,1, 1)

then yield

 

 

 

 

 

 

 

 

 

T = − Q111 (ks, 1, ki,1, 1) + Q113 (ks, 1, ki,1, 1) T 2

 

 

× Q131(ks, 1, ki,1, 1) + Q133(ks, 1, ki,1, 1)T 2

1 .

(2.94)

The expression of the transition matrix is identical to that given by (2.80), but the T 2 matrix is now given by (2.91) instead of (2.81). If the origins O1 and O2 coincide

Q311 (ki,1, 1, ki,1, 1) = −I

and we see that the T 2 matrix is the transition matrix of the inhomogeneity:

T 2 = T 2 = −Q112 (ki,1, 1, ki,2, 1) Q312 (ki,1, 1, ki,2, 1) 1 .

This formalism will be used in Sect. 2.5 to derive a recurrence relation for the transition matrix of a multilayered particle.


2.5 Layered Particles

115

2.5 Layered Particles

A layered particle is an inhomogeneous particle consisting of several consecutively enclosing surfaces Sl, l = 1, 2, . . . , N. Each surface Sl is defined with respect to a coordinate system Olxlylzl and we assume that the coordinate systems Olxlylzl have the same spatial orientation. The layered particle is immersed in a medium with optical constants εs and µs, while the relative media constants and the wave number in the domain between Sl and Sl+1 are εi,l, µi,l and ki,l, respectively. The geometry of a (multi)layered particle is shown in Fig. 2.4. The case N = 2 has been treated in the previous section and the objective of the present analysis is to extend the results established for two-layered particles to multilayered particles.

2.5.1 General Formulation

For a particle with N layers, the system of matrix equations consists in the null-field equations in the interior of S1,

Q133(ks, 1, ki,1, 1)i1 + Q131(ks, 1, ki,1, 1)i1 = −e ,

(2.95)

the null-field equations in the exterior of Sl−1 and the interior of Sl

 

−il−1 + Ql13(ki,l−1, l − 1, ki,l, l)il

 

+Ql11(ki,l−1, l − 1, ki,l, l)il = 0 ,

(2.96)

−Ql311(ki,l−1, l, ki,l−1, l − 1)il−1 + Ql33(ki,l−1, l, ki,l, l)il

 

+Q31

(k

, l, k

, l)i

l

= 0

(2.97)

l

i,l−1

i,l

 

 

 

l = 2, 3, . . . , N − 1

the null-field equations in the exterior of SN−1 and the interior of SN

Ds

 

S1

SN

Sl

Ol

Ol

 

O1

Ol-1

ON

Di,l Sl

 

Di,l-1 Sl-1

Di,l-2

Fig. 2.4. Geometry of a multilayered particle